User blog comment:Sarulu/Could we save jake?/@comment-32736638-20180126184121

Looker said at the end of the Hoopa fight that Jake would have a search party made for him, and he'll be fine.

So, there's not really a reason for any sort of theories when it's been confirmed that Jake's fine.

He'll probably show up after you beat the Champion like most Rivals in Pokemon Games do.